A planar model with multiple delays is studied. The singularities of the model and the corresponding bifurcations are investigated by using the standard dynamical results, center manifold theory and normal form method of retarded functional differential equations. It is shown that Bogdanov–Takens (BT) singularity for any time delays, and a serious of pitchfork and Hopf bifurcation can co-existent. The versal unfoldings of the normal forms at the BT singularity and the singularity of a pure imaginary and a zero eigenvalue are given, respectively. Numerical simulations have been provided to illustrate the theoretical predictions.
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References
Baldi P., Atiya A.F. (1994). How delays affect neural dynamics and learning. IEEE Trans. Neural Netw. 5(4): 612–621
Dieudonné J. (1969). Foundations of Modern Analysis. Academic Press, New York
Faria T. (2000). On a planar system modeling a neuron network with memory. J. Diff. Eq. 168, 129–149
Faria T., Magalhães L.T. (1995). Normal forms for retarded functional differential equations and applications to Bogdanov–Takens singularity. J. Diff. Eq. 122, 201–224
Giannkopoulos F., Zapp A. (2001). Bifurcations in a planar system of differential delay equations modeling neural activity. Phys. D 159, 215–232
Golubitsky M., Langford W.F. (1981). Classification and unfoldings of degenerate Hopf bifurcations. J. Diff. Eq. 41, 375–415
Guckenheimer J., Holmes P.J. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York
Kertesz V., Kooij R.E. (1991). Degenerate Hopf bifurcation in two dimensions. Nonlinear Anal. Theory Appl. 17(3): 267–283
Marcus C.M., Waugh F.R., Westervelt R.M. (1991). Nonlinear Dynamics and Stability of Analog Neural Networks. Phys. D 51, 234–247
Wei J., Li M.Y. (2004). Global existence of periodic solutions in a tri-neuron network model with delays. Phys. D 198, 106–119
Wei J., Velarde M.G. (2004). Bifurcation analysis and existence if periodic solutions in a simple neural network with delays. Chaos 14(3): 940–953
Wu J. (1998). Symmetric functional-differential equations and neural networks with memory. Trans. Am. Math. Soc. 350(12): 4799–4838
Yuan Y., Campbell S.A. (2004). Stability and synchronization of a ring of identical cells with delayed coupling. J. Dyn. Diff. Eq. 16(1): 709–744
Yuan Y., Wei J. (2006). Multiple bifurcation analysis in a neural network model with delays. Int. J. Bifurcations Chaos 16(10): 1–10
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Yuan, Y., Wei, J. Singularity Analysis on a Planar System with Multiple Delays. J Dyn Diff Equat 19, 437–456 (2007). https://doi.org/10.1007/s10884-006-9063-9
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DOI: https://doi.org/10.1007/s10884-006-9063-9